Since you're planning to backpack, try to find a trailhead with available permits that allows you to continue on to Half Dome. You can add a free Half Dome permit to a backcountry permit that is in the Half Dome vicinity.

But you may be looking at trying to get one of the First Come First Served Permits. Most of the reservable permits that head toward Half Dome are gone (Sunrise Lake > Cloud's Rest, Valley > Little Yosemite Valley, Glacier Point > LYV, and such)